Would I be in a position to specialize using GPA/class rank alone?
No, every specialty will want to see devotion to the chosen field through externship a, research, volunteer experience, etc. they want to know you know what you're getting into
Is research or anything else officially/unofficially mandatory for specializing?
See above. Research alone is not a requirement but you should have exposure to the field. I think ortho may appreciate research more than the other specialties but I'm not certain
What does it take to specialize in each of the different specialties?
The #1 thing for all of these specialties is showing you're interested and devoted. The following are also important factors
Omfs - grades, cbse score, externship
Peds - grades, externships
Ortho - grades, research, GRE score
Prosth - idk
Endo - grades and/or experience (most programs like completion of gpr or a few years pp)
perio - idk
Are there programs with stipends?
In omfs all programs pay a stipend, but for 6 year omfs you have to pay medical school tuition. Peds most pay a stipend, some not. Endo most do not pay you, a handful do. Same with prosth and ortho.
How strong of an influence does high school & undergraduate performance have on getting specialty program interviews and acceptances?
High school - not at all
College - very little, if at all, except for 6 year omfs programs where admissions to the medical
school may care about undergrad grades